OPTIMAL HEDGE RATIO

(The attached PDF file has better formatting.)

Study recommendation: Some exam problems test the optimal hedge ratio in varioushedges. Know the basic formula and its relation to CAPM betas and bond durations.

Exercise 1.2: Cocoa Hedges

A cocoa merchant with inventory of cocoa worth $10 million at present prices of $1,250 permetric ton is considering a risk-minimization hedge of the inventory using the cocoacontract of the Coffee, Cocoa, and Sugar Exchange.

! The volatility of returns for the cocoa inventory is 27%. ! The futures contract size is 10 metric tons. ! The volatility of the cocoa futures is 33%.! For the grade of cocoa in the inventory, the correlation between the change in the

futures price and the change in the spot cocoa price is 85%.

A. What is the risk-minimization hedge ratio?B. Should the cocoa merchant be long or short the futures contracts? C. How many contracts should be traded?

Part A: The optimal hedge ratio equals

SFthe correlation between the hedged good and the hedging instrument (D )

S× the standard deviation of the hedged good (F )

F÷ the standard deviation of the hedging instrument (F ), or

SF S Fhedge ratio = D × F / F

The optimal hedge ratio equals 0.85 × 0.27 / 0.33 = 0.6955.

Take heed: The volatility is the standard deviation times the square root of time. The ratioof volatilities equals the ratio of standard deviations, since the square root of time appearsin both numerator and denominator.

Part B: The cocoa merchant now owns the cocoa inventory. The merchant’s risk is that ahigher supply of cocoa or a lower demand for cocoa will reduce the price of cocoa. Themerchant must be short the cocoa futures contract.

Mnemonic: A party that owns an asset or commodity faces a risk that its price will decline.

! The long position pays a fixed price and gains if the price increases.

! The short position receives a fixed price and gains if the price decreases.

To hedge, the merchant must gain from the futures contract if the price declines, so themerchant takes a short position.

The cocoa merchant has goods to sell. The merchant is long the goods, so he or she mustsell futures contracts. A trader who has goods to sell must sell the futures contracts, anda trader who wishes to buy goods should buy the futures contracts.

Part C: If the hedge ratio were unity and the spot price and futures price had the samevolatility, one should trade futures contract just equal to the value of the hedged good, or$1,250 × 10 × Z = $10 million, or Z = 800 futures contracts. The optimal number ofcontracts to trade is 800 × 0.6955 = 556 contracts. Note that each contract is for 10 metrictons, and each metric ton is worth $1,250.

The optimal hedge ratio minimizes the variance of the total portfolio. If one buys " futurescontract for each unit of the underlying good of the same size, the total portfolio is

N goods + " × N futures.

The variance of this portfolio is

S S F FN × F + 2 × N × " × D × F × F + " × N × F 2 2 2 2 2 2

We set the partial derivative with respect to " equal to zero:

S F FN × [2 × D × F × F + 2" × F ] = 02 2

S F" = – D × F / F

The negative sign for " means that if the investor is long the commodity, he or she mustbe short the futures (if the correlation is positive), and if the investor is short the commodity,he or she must be long the futures (if the correlation is positive).

Exam Problems

For exam problems, the optimal hedge ratio comes in several guises.

A. Exam problems on commodity futures give standard deviations and correlations. Anexam problem might give the slope coefficient from a regression of the spot price onthe futures price. The slope coefficient is the optimal hedge ratio.

B. Exam problem on stock portfolios hedged with strike price futures give the beta of theportfolio, which is the optimal hedge ratio.

C. Exam problems on interest rate futures give the durations of the fixed-income securitiesportfolio and of the asset underlying the futures contract. The durations are like the

volatilities, and the correlation is one. The ratio of durations is the optimal hedge ratio.

Exercise 1.3: Cocoa and Poppy Hedges

A poppy farmer owns half of Afghanistan, with poppy farms that produce $150 million ofpoppy at present prices of $3,000 per metric ton. The volatility of returns for the poppy is51%.

Because of regime changes in Afghanistan, Iraq, and Turkey, the poppy farmer isconcerned about declines in the price of poppy. There is no poppy futures contract, butthe farmer knows that the price of poppies is inversely correlated with the price of cocoa.He is considering a risk-minimization hedge of the inventory using the cocoa contract ofthe Coffee, Cocoa, and Sugar Exchange. The contract size is 10 metric tons. The volatilityof the futures is 33%. The correlation between the cocoa futures and the poppy price is–45%. Compute the risk-minimization hedge ratio and determine the number of contractsto be traded.

Exercise 1.4: Beta Hedging

A stock portfolio of $100 million has a beta of 1.08 measured against the S&P futures,which stands at 350.00. How can one fully hedge this portfolio?

Solution 1.4: Full Hedging and Risk-Free Returns

Since we own the stocks now, we want to be short the futures contract. When the marketdeclines, the stock portfolio loses and a short futures contract gains. If the beta of thestock portfolio were unity, we would sell futures contracts for $100 million. Each S&P 500futures contract is 250 times the S&P 500 index, so we would sell

$100 million / (350.00 × 250) = 1,143 contracts.

Since the portfolio’s beta is 1.08, the optimal hedge ratio is 1.08, and we sell 1.08 × 1,143= 1,234 futures contracts.

Exercise 1.5: Virtual Stocks

A 20 year old investor inherits $50 million, but the estate will not settle for six months andhe will not receive the cash until that time. He believes that current stock values areattractive and he plans to invest in a diversified stock portfolio. How can he hedge thisanticipated investment using S&P futures?

Solution 1.5: Futures Hedging

The investor doesn’t own the stocks now, but expects to own them in the future. He isshort the stocks; to hedge against market value changes, he must be long the futurescontract.

A diversified stock portfolio has little non-systematic risk, so the correlation of the cashportfolio with the futures contract is close to unity. If an exam problem doesn’t state thecorrelation between the stock index price and the futures contract on the stock index,assume the correlation is unity.

Each futures contract is 250 × the value of the S&P 500 index, so he purchases $50 million/ (S&P index value × 250) futures contracts. He chooses futures contracts with a maturityof 6 months.

Question: What if the term is 5 months, and futures contracts trade with terms of 3 monthsand 6 months. Do we use a combination of both types of futures contract?

Answer: We use the futures contract that is longer than the optimal term. If we use the 3month futures contract, we have no hedge for the last two months. If we use the six monthfutures contract, we are hedged the whole five months, unless the futures price departsfrom the spot price.

OPTIMAL HEDGE RATIO

(Adapted from question 22 of CAS Exam 8, Spring 2002, Question 22, 2 points)

Exercise 1.6: Optimal hedge ratio

)S Change in spot price, S, during a period of time equal to the life of the hedge)F Change in futures price, F, during a period of time equal to the life of the hedge

SF Standard deviation of the change in the spot price

FF Standard deviation of the change in the futures priceh Hedge ratioD Coefficient of correlation between )F and )S

A. (0.5 point) If the trader is long the asset and short the futures contract, what is theexpression for the change in value of the hedged position over the life of the hedge?

B. (1.5 points) Define and solve for the optimal hedge ratio.

Trader’s Position

Part A: The trader holds one share and is short h units of the forward contract. Thehedged position is S – h × F, and the change in value of the hedged position is )S – h ×)F. The trader gets the change in the spot price, )S, minus (the change in the futuresprice, )F, times the hedge ratio h).

Part B: We minimize the variance of the hedged position, S – h × F, by setting the partialderivative with respect to h equal to 0.

S F S FThe variance is F + h × F – 2 × h × D × F × F2 2 2

Setting the partial derivative with respect to h equal to zero gives

F S F2h × F – 2 × D × F × F = 0,2

F Sor h = D × F / F .

Exercise 1.7: Hedging with Futures

(Adapted from question 32 CAS Spring 2003, Exam 8, 1.5 points)

A company has a $5 million portfolio with a beta of 1.3. It would like to use futurescontracts on the S&P 500 to hedge its risk. The index is currently at 815 and each contractis for delivery of $500 times the index.

1. How many futures contracts are needed to minimize the company’s risk?2. Should the firm be short or long the futures contract?3. (0.5 points) How many contracts should the company short to lower the beta of its

portfolio to 0.90?4. (0.5 points) What are two reasons a company would use futures contracts to hedge its

risk instead of selling the portfolio and investing the proceeds in Treasury bills?

Part A: The CAPM beta is the same as the optimal hedge ratio.

! If the portfolio had a beta of 1.0, the number of futures contracts is $5 million / ($500× 815) = 12.270.

! The portfolio has a beta of 1.3, so a $1 change in the index is a $1.30 change in theportfolio. The number of futures contracts is 12.270 × 1.3 = 15.951, or 16 contracts.

Take heed: Some candidates reason: “If a $1 change in the index is a $1.30 change in theportfolio, we need fewer futures contracts to get the same change in the portfolio. Why dowe enter into more futures contracts?”

The proper reasoning is as follows:

! A 1% decrease in the overall market price causes a 1% decrease in the index valueand a 1.3% decrease in the portfolio value.

! Suppose the portfolio’s value is the value of one index futures contract.! A 1% decline in the portfolio value is caused by a (1/1.3)% decline in the market, which

causes a (1/1.3)% decline in the futures price.! To make the decline in the futures price equal to the decline in the portfolio value, we

need 1.3 futures contracts.

Part B: The company owns the portfolio (that is, the company is long the portfolio), so itmust sell the futures contracts (or short the futures contracts) to hedge its risk.

Part C: To lower the beta of the portfolio to 0.9, the company must hedge (1.3 – 0.9) / 1.3= 30.77% of the risk. The number of futures contracts to short is 15.951 × 30.77% = 4.908,or 5 contracts.

Part D: If the company wants to change its investment strategy from a common stock

portfolio to a Treasury bill portfolio permanently, it should indeed sell the portfolio andinvest in Treasury bills. If the company wants to hedge, it has two possible motivations:

! It wants to eliminate the stock market risk for a short period, but to resume its commonstock investment strategy afterwards. If so, selling its common stock portfolio to buyTreasury bills and then selling the Treasury bills to buy common stocks is expensive.Using futures contracts to hedge the risk is cheaper (and quicker).

! The company may believe that it has superior stock selection expertise, so it wants toretain its superior portfolio of common stock, but either • it has average (or below average) stock market timing skills• it has superior stock market timing skills and it believes that now is a poor time to

invest in stocks.

The company uses futures contracts to eliminate the stock market timing risk whileretaining the superior common stock investment portfolio.

Take heed: Another common reason that firm hedge is to minimize their tax liabilities.

! Suppose the firm bought the stocks in 20X2, as the start of a bull market, for $3 million.! In 20X5, the stocks are worth $5 million, and firm wants to exit the market.! If it sells the stock, it owes a tax liability of $700,000.! If it hedges. It does not sell the stock, so it does not pay taxes on the capital gains.! Eventually, it sells the stock – perhaps 20 years later. Deferring the tax liability causes

a real increase in net wealth.

Exercise 1.8: S&P Index Futures Contracts

(Adapted from question 22 of the CAS Exam 8, Spring 2004)

A company owns a well-diversified portfolio of stocks worth $10 million with a beta of 1.50.

The company would like to use futures contracts on the S+P 500 index to change the betaof the portfolio to 2.00.

The S+P 500 index is currently 1,000 and the contract size is 250 times the index.

A. Should the company take a long or short position in the index futures contract?B. How many index futures should the company buy or sell?

Solution 1.8:

Part A: The firm now owns the portfolio (is long the stocks). To eliminate its market risk andminimize the variance of its total holdings (stock plus futures contracts), the firm must beshort the futures contracts.

To increase the sensitivity of its stock portfolio to market fluctuations, the company takesthe opposite position. It is speculating in stock risk, so it takes a long position in the indexfutures contracts.

Part B: The number of contracts is the change in beta times the ratio of the portfolio valueto the value of a contract:

number of contracts = (2.00 – 1.50) × $10 million / (1,000 × 250) = 20 contracts.

Take heed: The step-by-step guide uses $ – $N, where $ is the original beta of the stockportfolio and $ is the desired portfolio, giving –20 contracts.

! The Guide gives the number of short futures contracts.! –29 short position = 20 long position.

Solution #2: Alternatively, we evaluate the effect on the portfolio from a 1% change in themarket portfolio and determine how many futures contracts are needed.

! A 1% change in the market portfolio causes a 1.5 × 1% × $10 million = $150,000change in the portfolio.

! If the portfolio had a ß of 2, a 1% change in the market portfolio would cause a 2 × 1%× $10 million = $200,000 change in the portfolio.

! The company must buy futures contracts that have a $200,000 – $150,000 = $50,000value for a 1% change in the market portfolio.

! The futures contracts have betas of 1, so a 1% change in the market portfolio causes

a 1% × $1,000 × 250 = $2,500 change in the value of the futures contract.! The company needs $50,000 / $2,500 = 20 futures contracts.

Take heed: The firm is speculating on the market. If the firm thinks the market will moveup, it buys a high beta portfolio, which increases its gain from upward movements in themarket. Years ago, firms had to buy and sell shares to change the betas of the portfolio.The transaction costs of trading shares was so high that firms would not change theirportfolios unless they expected permanent shifts in the market. In contrast, buying andselling S+P index futures gives an immediate profit or loss and the costs of trading aresmall. The margins may be deposited as Treasury securities, so investors do not loseinterest on their margins.

The beta of a futures contract with no margin requirement is infinite because the initial costof the futures contract is zero, if we ignore the broker’s fees. We say the futures contractschange the beta of the portfolio. We might also say that the firm adds a stock portfolio witha $ of 1.500 to futures contract with a beta of +infinity to get a total portfolio with a beta of2.000

S&P 500 FUTURES CONTRACTS

Exercise 1.9: S&P 500 Futures Contracts

(Adapted from question 18 of the CAS Spring 2006 Exam 8 (1 point)

A firm has a stock portfolio with a beta of 0.75 and a value of $1,000,000.

The current level of the S&P 500 index is 1,000. One futures contract is for delivery of 250times the index. The futures contract have four months to maturity.

The firm wishes to hedge its portfolio over the next three months.

A. What risks can the firm hedge with S&P 500 futures?B. What risks can the firm not hedge with S&P 500 futures?C. Why would the firm hedge with S&P 500 futures instead of buying risk-free bonds?D. To hedge its portfolio against market risk, what transactions does the firm take? How

many futures contracts does it enter, and does it take a long or short position?E. To change its portfolio beta to 1.25, what transactions does the firm take? How many

futures contracts does it enter, and does it take a long or short position?F. Suppose the risk-free rate is 8% with quarterly compounding and the market rises 3%

over the next quarter (with quarterly compounding). For each futures contract above,show the changes in the portfolio and the return from the futures contracts.

Part A: The firm hedges the risk that the market may move down.

Part B: The firm can not hedge the risk that the stocks in its portfolio may do worse thanother stocks with similar betas.

Part C: Several types of firms may hedge with S&P 500 futures.

The firm may believe that the stocks in its portfolio are well chosen, with expected returnsabove those of other stocks with similar betas. However, the firm has no expertise inpredicting the direction of market movements, and it wants to minimize its overall stockvariance.

The firm may believe it has much expertise in predicting the direction of marketmovements. It speculates with S&P 500 futures by short and long positions.

The firm separately selects a desired level of market risk and the stocks it owns. It mayselect a desired $ of 1.25 and then select individual stocks. Before trading of S&P 500futures, it had to select stocks so that the average $ is 1.25. With S&P 500 futures, itselects the stocks with the highest abnormal returns and uses S&P 500 futures to adjustthe beta.

Part D: The optimal hedge ratio is the beta of 0.750.

! Each futures contract is worth $1,000 × 250 = $250,000.! The firm now owns the stocks, so it uses a short position in the S&P 500 futures.

The firm takes a short position in 0.75 × $1 million / $250,000 = 3 futures contracts.

Part E: The firm is speculating with S&P 500 futures, not hedging. To increase its marketrisk, it takes a long position in the futures contracts. It wants to increase its beta to 1.25from 0.75, so its added beta = its optimal hedge ratio = 1.25 – 0.75 = 0.50.

It takes a long position in 0.50 × $1 million / $250,000 = 3 futures contracts.

Part F: The S&P 500 index has a beta of 1. The risk-free rate over one quarter is ¼ × 8%= 2%. The 3% return on the market is a 1% excess return.

! The market’s excess return is 1%, so the firm’s portfolio rises by $1 million × (2% + 0.75× 1%) = $27,500.

! The firm wants the portfolio’s value to rise by $1 million × (2% + 1.25 × 1%) = $32,500.

! It buys strike price futures contracts that rise (in aggregate) by $5,000 if the marketrises 1%.

! If the market rises 1%, the S+P futures contract rises by $250,000 × 1% = $2,500.

! The firm should buy 2 futures contracts.

Jacob: What about the 3 month hedge vs the 4 month futures contract? How does thisaffect the solution:

Rachel: The firm should buy a futures contract with shortest maturity that equals orexceeds the term of the hedge. After three months, it sells the futures contracts. Theslight mis-match adds a small basis risk, but this is not significant.